By analyzing the eigenvalues and eigenvectors of the
covariance matrix derived from the points on the shapes, we can
determine that the above shapes are unstable.
Below each figure
is the number and type of instabilities. The axis
of unstable rotation and direction of unstable translation
are indicated on the shapes.

Points picked by our algorithm to constrain the
unstable motions of a patch with two grooves. (a) Points
constraining two unstable translational eigenvectors. (b)
Points constraining unstable rotation. (c)-(d) Two remaining
rotations are stable so they only require a few points. The
eigenvector corresponding to translation in z is well
constrained by the already picked points and does not
contribute to the sampling.

Abstract

The Iterative Closest Point (ICP) algorithm is a widely used method
for aligning three-dimensional point sets. The quality of alignment
obtained by this algorithm depends heavily on choosing good pairs of
corresponding points in the two datasets. If too many points are
chosen from featureless regions of the data, the algorithm converges
slowly, finds the wrong pose, or even diverges, especially in the
presence of noise or miscalibration in the input data. In this paper,
we describe a
method for detecting uncertainty in pose, and we propose a point
selection strategy for ICP that minimizes this uncertainty by choosing
samples that constrain potential unstable transformations.